# Questions about log polar transform

I read various papers about the log polar transform and its application on template matching with images and have some questions:

1. In "Image Registration Using Log Polar Transform and Phase Correlation to Recover Higher Scale" the authors say: "In this algorithm first the sense image is downscaled by the factor of 2. Log-Polar algorithm gives best result for any value of rotation and translation but it will not work if the scale is more than 1.8 or 2, as the frequency components alters at high scale [7]." In the referenced paper "IMAGE REGISTRATION USING LOG-POLAR TRANSFORM AND PHASE CORRELATION" the authors only mention at the end that their results are verified with a scaling factor up to 2.0, there is no mention at all about altering frequency components at high scale. Then I read through http://cvpr.uni-muenster.de/teaching/ss08/seminarSS08/downloads/Wentker-Vortrag.pdf . He says that the polar transform is unable to detect scales > 2, the log transform doesn't have it. This clearly contradicts the first paper, which says the log polar transform can only go to scales < 2. So does the log polar transform normally only work up to scale 2.0?

2. Regarding question 1, what exactly is the reason that it only works up to a scale of 2? The altering frequency components are mentioned, but I don't know what they really mean with that and it is never explained in any of the papers I found, only that this value exists.

3. In the presentation http://cvpr.uni-muenster.de/teaching/ss08/seminarSS08/downloads/Wentker-Vortrag.pdf page 22, we have the normal, polar and log-polar transformed image. The polar image has the black areas, which are parts that can't be mapped to the original image because it would be outside of its boundaries. However, these areas have nearly disappeared in the log polar image. Why is that? I tested various Logpolar implementations using OpenCV, Python, another example is Scale and Rotation invariant Template Matching and all the time, the log polar images looks more like the middle image and not like the right one. What exactly is happening here? Why does the third image look so different, compared to other log polar images which look more like the second?

I hope you can answer my questions.